Importance of Correct Representation-Problem Solving

Successful problem solving depends on representing problems in such a way that appropriate operators can be seen to apply.

We have analyzed a problem solution as consisting of problem states and operators for changing states. So far, we have discussed problem solving as if the only tasks involved were to acquire operators and select the appropri- ate ones. However, there are also important effects of how one represents the problem. A famous example illustrating the importance of representation is the mutilated-checkerboard problem (Kaplan & Simon, 1990). Suppose we have a checkerboard from which two diagonally opposite corner squares have been cut out, leaving 62 squares,  Now suppose that we have 31 dominoes, each of which covers exactly two squares of the board. Can you find some way of arranging these 31 dominoes on the board so that they cover all 62 squares? If it can be done, explain how. If it cannot be done, prove that it cannot. Per- haps you would like to ponder this problem before reading on. Relatively few people are able to solve it without some hints, and very few see the answer quickly.
The answer is that the dominoes cannot cover the check- erboard. The trick to seeing this is to include in your repre- sentation of the problem the fact that each domino must cover one black and one white square, not just any two squares. There is just no way to place a domino on two squares of the checkerboard without having it cover one black and one white square. So with 31 dominoes, we can cover 31 black squares and 31 white squares. But the mu- tilation has removed two white squares. Thus, there are 30 white squares and 32 black squares. It follows that the mutilated checkerboard cannot be covered by 31 dominoes.
Contrast this problem with the following “marriage” problem that occurs with many variations in its statement:
In a village in Eastern Europe lived an old marriage broker. He was worried. Tomorrow was St. Valentine’s Day, the village’s traditional betrothal day, and his job was to arrange weddings for all the village’s eligible young people. There were 32 women and 32 young men in the village. This morning he learned that two of the young women had run away to the big city to found a company to build phone apps. Was he going to be able to get all the young folk paired off?
People almost immediately see that this problem cannot be solved since there are no longer enough women to pair up with the men.3
Since both problems require the same insight of matching pairs (black with white squares in the case of the checkerboard, and men with women in the case of marriage), why is the mutilated-checkerboard problem so hard and the marriage problem so easy? The answer is that we tend not to represent the checkerboard in terms of matching black and white squares whereas we do tend to represent mar- riages in terms of matching brides and grooms. If we use such a matching repre- sentation, it allows the critical operator to apply (i.e., checking for parity).
Another problem that depends on correct representation is the 27-apples problem. Imagine 27 apples packed together in a crate 3 apples high, 3 apples wide, and 3 apples deep. A worm is in the center apple. Its life’s ambition is to eat its way through all the apples in the crate, but it does not want to waste time by visiting any apple twice. The worm can move from apple to apple only by go- ing from the side of one into the side of another. This means it can move only into the apples directly above, below, or beside it. It cannot move diagonally. Can you find some path by which the worm, starting from the center apple, can reach all the apples without going through any apple twice? If not, can you prove it is impossible? The solution is left to you.
Inappropriate problem representations often cause students to fail to solve problems even though they have been taught the appropriate knowledge. This fact often frustrates teachers. Bassok (1990) and Bassok and Holyoak (1989) studied high-school students who had learned to solve such physics problems as the following:
What is the acceleration (increase in speed each second) of a train, if its speed increases uniformly from 15 m/s at the beginning of the 1st second, to 45 m/s at the end of the 12th second?
Students were taught such physics problems and became very effective at solv- ing them. However, they had very little success in transferring that knowledge to solving such algebra problems as this one:
Juanita went to work as a teller in a bank at a salary of $12,400 per year and received constant yearly increases, coming up with a $16,000 sal- ary during her 13th year of work. What was her yearly salary increase?

The students failed to see that their experience with the physics problems was relevant to solving such algebra problems, which actually have the same struc- ture. This happened because students did not appreciate that knowledge associ- ated with continuous quantities such as speed (m/s) was relevant to problems posed in terms of discrete quantities such as dollars.
■ Successful problem solving depends on representing problems in such a way that appropriate operators can be seen to apply.